Presentation Embedding--Another Case of Stumbling Progress

EMBEDDING: ANOTHER CASE OF STUMBLING PROGRESS

Jens Høyrup [email protected] http://ruc.dk/~jensh/

Contribution to the workshop Mathematics in the Renaissance Language, Methods, and Practices ETSEIB, Barcelona, 23 January 2015

1 Approaches to algebraic symbolism

Georg Nesselmann, Algebra der Griechen (1842): a classiﬁcation of algebra types into three groups: rhetorical, syncopated, and symbolic.

In “rhetorical algebra”, everything in the calculation is explained in full words.

“Syncopated algebra” uses standard abbreviations for certain recurrent concepts and operations, while “its exposition remains essentially rhetorical”.

In “Symbolic algebra”, “all forms and operations that appear are represented in a fully developed language of signs that is completely independent of the oral exposition”.

2 Though characterized as “stages” (Stufen), clear from his examples that this division into chronological stages is at most meant locally, not as steps of universal history.

The rhetorical stage is represented by Iamblichos, “all so far known Arabic and Persian algebraists”, and all Christian-European writers on algebra until Regiomontanus.

3 Diophantos,and the later Europeans until well into the 17th century are classiﬁed as syncopated, although already Viète has sown the seeds of modern algebra in his writings, which however only sprouted some time after him.

In the following pages, Oughtred, Descartes, Harriot and Wallis are mentioned as founders of symbolic algebra.

However, we Europeans since the 17th century are not the ﬁrst to have attained [the symbolic] level; indeed, the Indian mathematicians anticipate us in this domain by many centuries.

4 The typology is still in common use by historians who refuse to see every use of algebraic abbreviations as a “symbolism”,

For the sane reason, it is often criticized – obvious mistakes from the 1840s would have been forgotten long ago. (In the following, I shall formulate some supplementary objections)

5 However, the central observation about what characterizes the symbolic level has mostly been overlooked – namely that symbolization allows operations directly on the level of the symbols, without any recourse to thought through spoken or internalized language.

In Nesselmann’s own words We may execute an algebraic calculation from the beginning to the end in fully intelligible way without using one written word, and at least in simpler calculations we only now and then insert a conjunction between the formulae so as to spare the reader the labour of searching and reading back by indicating the connection between the formula and what precedes and what follows.

6 That is what we do when we reduce an equation by additions, divisions, differentiations, etc.

We can of course speak about the operations we perform, just as we may speak about the operations we perform when changing the tyre on a bicycle or preparing a sauce; but in all three cases the operations themselves are outside language.

7 As illustration, two instances of incipient symbolic operation – one from Diophantos, one from the Italian 14th century.

In the Arithmetic, Diophantos uses abbreviations (spoken of as “signs” [σημειον]) for the unknown number (the arithmós) and its powers.

The unknown arithmós itself is written ς; for the higher powers (dynamis = ς2, kybos = ς3, dynamodynamis = ς4, etc.), phonetic complements are added (ΔΥ, ΚΥ, etc.).

Similarly, complements are added to the sign for the monad (“power zero”), and for numbers occurring as denominators in fractions, except in the compact writing of fractions where“ 5 ” means16 . 16 5

8 Addition is implied by juxtaposition, subtraction and subtractivity are denoted by the abbreviation (λειψις, “missing” etc.).

Only one sign occasionally serves direct operation: the designation of the “part denominated by” n (better, since n is not always integer, the reciprocal of n);

In III.xi a number which was posited to be ς× is stated immediately to be“41 ” (77 ) when ς itself turns out to be“77 ” . 77 41 41

This would hardly have been possible if Diophantos had not known at the level of symbols (and supposed his reader to recognize) that ς× × ς p × q ( ) = , and that “q ” =“p ” . This may be the only instance of genuine symbolic operation.

9 Next: a Tuscan Trattato dell’alcibra amuchabile, a compound in three parts from c. 1365.

In the third part the request to divide 100 ﬁrst by a “quantity” and then by the “quantity” plus ﬁve. The sum of the quotients is told to be 20.

So, you should ﬁrst divide 100 by a cosa (“a thing”), and next by “a cosa and 5”, and join the two quotients.

Similar problems (though with subtraction) are found in al- Khwa¯rizmı¯’s and Abu¯Ka¯mil’s algebras and again in Fibonacci’s Liber abbaci.

10 Al-Khwa¯rizmı¯ gives a purely numerical (but sensible) prescription for the initial, difficult steps obviously, what he did went beyond his technical vocabulary.

Abu¯Ka¯mil uses a geometric diagram; and Fibonacci applies proportions.

11 The fourteenth-century treatise, however, comes close to what we would do:

Now I want to show you something similar so that you may well understand this addition, and I shall say thus: I want to join 24 divided by 4 to 24 divided by 6, and you see that it should make 10. Therefore 24 write 24 divided by 4 as a fractions, from which comes /4. And posit similarly 24 divided by 6 as a fractions. Now multiply in cross, that is, 6 times 24, it makes 144; and now multiply 4 times 24, which is above the 6, it makes 96, join it with 144, it makes 240. Now multiply that which is below the strokes, that is, 4 times 6, it makes 25. Now you should divide 240 by 240, from which 10 should result. [...]

12 Then follows the application: Now let us return to our problem. Let us take 100 divided by a cosa and 100 divided by a cosa and 5 more, and therefore posit these two divisions as if they were fractions, as you see hereby.

And now multiply in cross as we did before, that is, 100 times a cosa, which makes 100 cose. And now multiply along the other diagonal, that is, 100 times a cosa and ﬁve, it makes 100 cose and 500 in number; join with 100 cose, you get 200 cose and 500 in number. Now multiply that which is below the strokes, one with the other, it makes a censo and 5 cose more. Now multiply the results, that is, 20 against a censo and 5 cose more, it makes 20 censi and 100 things more, which quantity equals 200 things and 500 in number. ...

13 This text, we see, is purely rhetorical – everything is written out in full words.

On the other hand, the solution proceeds by means of formal operations, in a way we are accustomed to in symbolic algebra; rhetorically expressed polynomials are dealt with as if they were the numbers of normal fraction arithmetic.

We may say that the lexicon of the text is rhetorical, but its syntax (in part) symbolic.

14 Characteristic for this syntax is the phenomenon of embedding: the insertion of something possibly complex in the place of something simpler.

We know the phenomenon from ordinary language making use of subordinate clauses: Igonow→ I go when it pleases me.

15 In contemporary symbolic mathematics indeﬁnitely nested embedding is possible – for instance, in continued fractions, or in the graphically simpler expression x x x x x 1+ /2 (1 + /3 (1 + /4 (1 + /5 (1 + /6 (...)))) .

In ordinary language, the same possibility is present, restricted only by pragmatic considerations of comprehensibility

“This is the man all tattered and torn / That kissed the maiden all forlorn / That milked the cow with the crumpled horn / That tossed the dog / That worried the cat / That chased the rat / That ate the cheese / That lay in the house that Jack built”.

16 We may now turn back to Nesselmann.

He ascribed to the Indian mathematicians a symbolic algebra that precedes that of Europe by many centuries.

17 An example, borrowed from Bhaskara II (b. 1115)

What we would express 5x+8y+7z+90=7x+9y+6z+62 is written by Bha¯skara as a scheme yâ 5 kâ 8 nî 7 rû 90 yâ 7 kâ 9 nî 6 rû 62 while our 8x3+4x2+10y2x =4x3+0x2+12y2x appears as yâ gha 8 yâ va 4 kâ va yâ.bhâ 10 yâ gha 4 yâ va 0 kâ va yâ.bha 12

18 Datta and Singh quote David Eugene Smith for the stance that this notation is “in one respect [...] the best that has ever been suggested”, namely because it “shows at a glance the similar terms one above the other, and permits of easy transposition”.

However, the Indian schemes do not permit direct multiple embedding – for instance the replacement of yâ by a polynomial.

Such an operation requires the same amount of thinking in the Indian notation as in a rhetorically expressed algebra.

19 Not impossible in either case: al-Khwa¯rizmı¯’s solution of the above-mentioned problem shows that he, or somebody who inspired him, could do something of the kind; but al-Khwa¯rizmı¯ did not have the words to explain what he was doing (as he is generally fond of), free formal operations were only made possible by the fully developed symbolism of an Euler.

20 Indian schemes allow direct operations, and in this sense they clearly constitute a symbolism, as claimed by Nesselmann.

However, Smith is right that this notation is the best “in one respect” only – namely within the restricted framework of problems actually dealt with by Bha¯skara. It allows operations directly at the level of symbols, but only a limited, non-expandable set of operations.

21 Stumbling progress toward algebraic symbolism

On an earlier occasion I have described the slow development of algebraic symbolism, from the ﬁrst introduction in late twelfth- century Maghreb to the ﬁnal unfolding around Viète and Descartes – not only “hesitating”, as my title said, but stumbling.

I shall give a short summary.

At some moment mathematicians in the Islamic West (the Maghreb, in the general sense including also al-Andalus) invented not only the writing of fractions with a stroke (taken over in the Latin Liber mahameleth, plausibly from the 1160s) but also notations for composite fractions, most important the notation for ascending continued fractions such aseca meaninga +c +e (also used by Fibonacci) fdb b bd bdf

22 Probably towards the very end of the century (Fibonacci does not know about it), an algebraic symbolism was created, with symbols for powers zero to three of the unknown, and signs for subtraction, inverse, square root and equality; ibn al-Ya¯samin may have been the inventor.

First described by Franz Woepcke on the basis of its use by al- Qalasa¯dı¯ (15th c.) in 1854, that is, well after Nesselmann’s perspicacious reference to “all so far known Arabic and Persian algebraists”.

Already Woepcke suspected from ibn Khaldu¯n’s report that the notation might go back to the twelfth century, as now conﬁrmed.

23 Signs for the powers are written above their coefficient, the root and inverse signs above their argument.

The signs are derived from the initial letters of the corresponding words but provided with tails enabling them to cover composite expressions, that is, to delimit “algebraic parentheses”

The notation served to write polynomials and equations, and even to operate on the equations.

24 What is “a parenthesis”, and what is an “algebraic parenthesis”?

A parenthesis is not a (round, square or curly) bracket nor a pair of brackets but an expression that is marked off, for example by a pair of brackets;

but in spoken language, pauses may mark off a parenthesis in the ﬂow of words, and in written prose this is often delimited by a pair of dashes.

25 An algebraic parenthesis marks off a single entity which can be submitted to operations as a whole;

– in calculations it has to be determined ﬁrst.

– When division is indicated by a fraction line, this line delimits the numerator as well as the denominator as parentheses if they happen to be composite expressions (for instance, polynomials).

– Similarly, the modern root sign marks off the radicand as a parenthesis.

26 The early Maghreb evidence is accidental, but later extant writings are sometimes systematic in their use of the notation, showing that at least its fully developed form can be regarded as a genuine symbolism at the Indian level (though so different in character that inﬂuence one way or the other can be safely disregarded).

In these writings, the symbolic calculations are as a rule made separate from the running text, usually following after a phrase “its image is” and thus illustrating the rhetorical exposition.

They can also stand as marginal commentaries. Such marginal calculations probably correspond to what was on a takht (a dustboard, in particular used for calculation with Hindu numerals) or a lawha (a clayboard used for temporary writing)

27 Fibonacci does not know the Maghreb notation (his copious use in non-algebraic contexts of rectangular schemes rendering what would be written on a lawha makes it almost certain he would have used it if he had known about it).

Nor does the earliest generation of abbacus algebra as represented by Jacopo da Firenze’s Tractatus algorismi.

Even algebraic abbreviations are absent in this earliest phase, even though abbreviations are of course used in the writing of current words.

28 Soon, however, some traces of symbolic operation turn up.

Paolo Gherardi’s Libro de ragioni from 1328 describes operations on a diagram (itself missing in the copy, which also has a defective text on this point, but present in a parallel text):

The context once more the problem about dividing 100 ﬁrst by a quantity, then by 5 more, and adding the two quotients.

Clearly, the same operations are thought of, even though the diagram is more rudimentary.

29 In the ﬁrst part of the Trattato dell’alcibra amuchabile, schemes are used to teach the multiplication of binomials – for example (observe the abbreviation for radice, “root”):

The binomials are numerical, but since al-Khwa¯rizmı¯ irrational roots had been used so to speak as pedagogical stand-ins for algebraic roots (square roots of the censo, that is, cose).

30 The Trattato dell’alcibra amuchabile was written in c. 1365, but this part of its material is probably older (as are others demonstrably).

In Dardi of Pisa’s Aliabraa argibra from 1344, we ﬁnd something similar though more elaborate:

Here we ﬁnd a supplementary abbreviation, for meno, “less”.

31 Dardi indeed uses abbreviations systematically: radice is always , meno (“less”) is , cosa is c, censo is Ç, numero/numeri are n˜uo/n˜ui. Cubo is unabridged, censo de censo (the fourth power) appears not as ÇÇ but as ÇdeÇ(an expanded linguistic form which we may take as an indication that Dardi merely thinks in terms of abbreviation and nothing more).

Roots of composite entities are written by a partially rhetorical expression, for instance “ de zonto1 cô de 12” 4

1 (meaning cosa 12 ; zonto means “joined”; that a root is “joined” 4 means that it is taken of a binomial or other composite expression). NB: A parenthesis, though often badly delimited

32 Algebraic monomials are written in a way that might suggest an inversion of the Maghreb notation – for instance, “4 cose” is written4 . c

The same notation is used in the original manuscript of a Trattato di tutta l’arte dell’abbacho from 1334.

Closer inspection of the use shows that the notation must be understood as a reﬂection of the spoken form, in analogy with the frequent writing of il terzo (“the third”, also when number 3 in a sequence is meant) as “il1 ” 3

33 That is, the fraction notation itself is not understood as a division but as an indication of the ordinal form of the numeral. Maghreb inspiration, if present, has not been accompanied by understanding.

Even though Dardi was indubitably the best abbacus mathematician of his days – and the ﬁrst to write a treatise dealing solely with algebra, – and more consistent in his use of algebraic standard abbreviations than anybody else in his century, he saw no point in exploring the possibilities of symbolic operations.

34 All in all, until the mid-14th century: the only symbolic operations we ﬁnd are those on formal fractions and the multiplication of binomials in schemes – both rather rudimentary, the former plausibly inspired from Maghreb practices, the latter possibly an independent development.

Algebraic abbreviations remained abbreviations and nothing more, and only Dardi used them systematically.

35 In the early 15th century, the use of standard abbreviations (co and ce) for cosa and censo become common (but more often used in marginal annotations than in the running text, rarely very systematic, and very rarely used for symbolic operations); they are often written above the coefficient, which might suggest inspiration from Maghreb ways.

The ﬁrst trace of such recent interaction (supported by textual details) is the algebra section of a Tratato sopra l’arte della arismetricha written in Florence around 1390.

Beyond sophisticated use of polynomial algebra in the transformation of equation types, we ﬁnd here a clear discussion of the sequence of algebraic powers as a geometric progression.

36 The running text contains no abbreviations and certainly nothing foreshadowing symbolic operations.

Inserted to the left, however, we ﬁnd a number of schemes explained by the text and showing multiplication of polynomials with two or three terms (numbers, roots and/or algebraic powers).

Those involving only binomials are related to those of the Trattato dell’alcibra amuchabile and Dardi.

The schemes for the multiplication of three-term polynomials are of a different kind. They emulate the scheme for multiplying multi-digit numbers, and the text itself justly refers to multiplication a chasella, very close to what is found in Maghreb manuscripts.

37 Such schemes (and other schemes for calculation with polynomials) turn up not only in later abbacus writings (for instance, in Raffaello Canacci’s Ragionamenti d’algebra, on which more presently) but also in Stifel’s Arithmetica integra (1544), Jacques Peletier’s L’Algèbre (1554) and Petrus Ramus’s Algebra (1560).

On the other hand, schemes of this type are absent from the three major “abbacus encyclopediae” from c. 1460, all three Florentine and in the tradition reaching back via Antonio de’ Mazzinghi (c. 1353 to c. 1391 to Paolo dell’Abbaco and Biagio “il vecchio” (mid- and early- mid-14th c.).

38 Most famous and known from many copies is Benedetto da Firenze’s Trattato de praticha d’arismetrica.

The other two are Florence, Palatino 573, and Vatican, Ottobon. lat. 3307 – the compilers of the latter two being both pupils of a certain Domenico d’Agostino vaiaio.

39 On the other hand, in these we ﬁnd marginal schemes of this type:

Antonio marginalberegning

A marginal calculation accompanying the same problem from Antonio’s Fioretti in Siena L.IV.21, fol. 456r and Ottobon. lat. 3307, fol. 338v.

The appearance of the scheme in similar shape in the different encyclopediae suggests that it goes back to Antonio (from whom the problem itself is borrowed). 40 We also ﬁnd an abundance of formal fractions, and schemes of a different kind for the multiplication of binomials (ρ stands for cosa, c for censo):

Benedetto 455r

41 All in all, as I summarized the matter 2010:

The three encyclopediae confirm that no systematic effort to develop notations or to extend the range of symbolic calculation characterizes the mid- century Italian abbacus environment – not even among those masters who, like Benedetto and the compiler of Palat. 573, reveal scholarly and Humanist ambitions [...]. The experiments and innovations of the fourteenth century – mostly, so it seems, vague reflections of Maghreb practices – had not been developed further. In that respect, their attitude is not too far from that of mid-fifteenth-century mainstream Humanism.

As Humanism, the character and use of notations underwent some changes toward the end of the century – and not only because of the generalization of printing. (the notational innovations are also found in manuscripts, and sometimes they are more thorough there).

42 Firstly, the use of abbreviations becomes more systematic, and there is some exploration of alternative systems; secondly, the character of the sequence of powers as a geometric series is taken note of more often, and the sequence of powers is linked to the natural numbers.

Sometimes the numbering coincides with our exponents, but the most inﬂuential work – Luca Pacioli’s Summa – makes the unfortunate choice to count number as level 1, and cosa as level 2 (etc.). In consequence, it still asks for thinking to see that an equation involving (for example) censi di censo, censi and numero is simply a quadratic equation with unknown censo.

43 We still ﬁnd schemes for multiplication of binomials, sometimes like those of Dardi, sometimes similar to Benedetto’s, and also symbolic marginal calculations similar to what Benedetto and his contemporaries had offered – but hardly anything that goes beyond them.

We may jump – in time, and also socially, namely to a scholar treating in Latin of abbacus mathematics von höheren Standpunkt aus – to Cardano’s Practica arithmetice, et mensurandi singularis from 1539.

44 Here, we ﬁnd not only indented marginal schemes (in Benedetto style) but also compact writings in the running texts – a very simple case is the statement (C viir) that ducendo .8 ad . ﬁt 64; somewhat more complex (D ir) 1.co.p. ˜ 1.men.1.co. , meaning 1 cosa + 1–1cosa . 1.ce.piu.1. 1censo 1 “Plus”, we observe, may appear both asp ˜ and as piu. As we shall see, the use of the parenthesis function is even less systematic in Cardano’s Ars magna from 1545.

It is doubtful whether this can have assisted symbolic operations, and even whether it has supported thought better than full writing (as the marginal schemes indubitably do, but only for the addition, subtraction and multiplication of binomials, which they had always served).

45 Tartaglia’s Sesta parte del general trattato from 1560 is not very different in its use of notations: there are schemes for the operations on binomials, still in Benedetto’s style. (trinomials are treated stepwise, the a casella scheme for polynomials from the Tratato sopra l’arte della arismetricha seem to have been forgotten).

We also ﬁnd formal fractions like240ce. men 48000 and other expressions 1ce. p 14 co. p 60 using abbreviation used in the running text – but nothing with suggests thought supported by symbolic operations.

46 Michael Stifel, in the Arithmetica integra (1544), as already Christoph Rudolff in the Coss [1525], use the modern symbols +, – and √, but without making any other changes.

Noteworthy innovations are to be found in the works of Chuquet and Bombelli, but since these innovations are central to my topic I shall deal with them below.

47 Powers

I shall now return, not to embedding as a mere fact but to the willingness to think in terms of embedding.

This willingness is revealed by the ways in which higher powers were named.

48 Diophantos introduces these terms for the powers of the unknown (telling them to “having been approved”, that is, to be already established) αριθμóς (ﬁrst power) δυναμις (second power) κυβος (third power) δυναμοδυναμις (fourth power) δυναμοκυβος (ﬁfth power) κυβοκυβος (sixth power)

Obviously, juxtaposition here means multiplication. Nothing in the grammatical construction would suggest otherwise, the nouns are glued together in the standard way to make compositions.

49 Arabic algebra is very similar. A systematic exposition was given by al-Karajı¯ in the Fakhri: jidhr or šai (first power) ma¯l (second power) ka b (third power) ma¯l ma¯l (fourth power) ma¯l ka b (fifth power) ka b ka b (sixth power) ma¯l ma¯l ka b (seventh power) ka b ka b (eighth power) ka b ka b ka b (ninth power) “and so on, until infinity”; indeed, the system allows naming of all powers.

50 Juxtaposition once more stands for multiplication. Grammatically, the connection between the nouns is a genitive, but the Semitic genitive does not, like its Indo-European namesake, necessarily imply a subordination as we shall see, the use of the Latin and Italian genitive was in the long run to enforce a reading of “the cube of the cube” as the ninth power.

51 Only in the long run, however.

The Latin translations of al-Khwa¯rizmı¯’s algebra have no names for powers beyond the second.

The Liber mahameleth refers twice to the cubus, explains the ﬁrst time that the cubus is the product of the census and its root but goes no further in the sequence.

52 Fibonacci, however, does. He does not explain the names nor a fortiori the whole sequence, but in the Liber abbaci he makes use of those which he needs.

His sixth power may be cubus cubi as well as census census census (this equivalence is stated, while the eighth power is census census census census.

He obviously uses the Latin genitive in the Arabic way.

53 The early abbacus algebras – for instance, Jacopo – go no further than the fourth power, which is censo di censo (while the third power is cubo). They can thus tell us nothing about conceptualizations.

The earliest abbacus writer known to go beyond this limit is a certain Giovanni di Davizzo in 1339.

The interesting part for our present discussion ﬁrst gives rules for the multiplication of powers, some of which show the thinking to be multiplicative in spite what might be suggested by the grammar: and thing times censo makes cube and cube times cube makes cube of cube and censo times cube makes censo of cube.

54 Then follows something which will wring the bowels of any modern mathematician – a daring but mistaken attempt to express negative powers, namely confounding them with roots (the ﬁrst negative power stated to be “number”): And know that dividing number by thing gives number and dividing number by censo gives root and dividing thing by censo gives number and dividing number by cube gives cube root and dividing thing by cube gives root and dividing censo by cube gives number and dividing number by censo of censo gives root of root and dividing thing by censo of censo gives cube root and dividing censo by censo of censo gives root and dividing cube by censo of censo gives number and dividing number by cube of cube gives cube root of cube root and dividing thing by cube of cube gives root of cube root and dividing censo of censo by cube of cube gives root ...

55 It may not be warranted to take the text as more than a play with words, but we can still try to take it as seriously meant, and suppose Giovanni’s “roots” in this context to be the same as those he speaks about in the ﬁrst part of the excerpt.

Under these conditions we see that even his roots are supposed to be composed “multiplicatively” – for instance, that the cube root of the cube root is the sixth, not the ninth root.

Similarly, the Trattato dell’alcibra amuchabile takes radicie de radicie chubica to be the ﬁfth – but since this is once more in a messy context where no calculation is performed, the compiler has no reason to discover that his rule is absurd.

56 Dardi knows better. His names for the powers are still in the Arabic style, and he even explains like Fibonacci that ÇdiÇdiÇis the same as cubi di cubi.

Since most of his problems involve radicals (in the style of “roots of cubes”), he gives us the occasion to observe that he is aware that repeated root taking involves embedding – expressing for example the twelfth root as cuba de de overo de

3 3 de cuba (a ora ), while his term for the twelfth power would be cubo di cubo di cubo di cubo.

57 But this terminological insight and innovation has a price: Dardi has no name for the ﬁfth and the seventh root, and once replaces the former by cuba and once the latter by dela ,, thus ending up with mistaken rules.

In spite of his manifest command of the sequence of powers, he is at the limits of what he can express.

58 Toward 1400, the frontier has moved, and the consequences of the genitive construction are made themselves felt – but as yet inconsistently.

The manuscript Palat. 573 (one of the three “abbacus encyclopedia” mentioned above) quotes Antonio de’ Mazzinghi for the following: Cosa is here a hidden quantity; censo is the square of the said cosa; cubo is the multiplication of the cosa in the censo; censo di censo is the square of the censo, or the multiplication of the cosa in the cubo. And observe that the terms of algebra are all in continued proportion; such as: cosa, censo, cubo, censo di censo, cubo relato, cubo di cubo, etc.

Antonio avoids speaking of the fifth power as cubo di censo or censo di cubo, replacing it be a neologism; but his naming of the sixth power is still multiplicative. The name for the fifth power may have been inspired by his term for the fifth root, radice relata.

59 The algebra section of a Tratato sopra l’arte della arismetricha (Florence, c. 1390, we remember) – also from the hand of a highly competent algebraist – starts by explaining how the powers are produced one from the other, and that they are in continued proportion.

One particularity is an extra identification of these as “roots”, namely (as explained) as the roots which they have. The sequence is cosa (first power) censo or radice (second power) cubo or radice cubica (third power) censo di censo or radice della radice (fourth power) cubo di censi or “a root born from a squared root multiplied against a cubicated quantity” or (some say) radice relata (fifth power) censo di cubo (sixth power)

60 For the sixth power it is stated (but not given as a name) that one may take the root, and of this quantity take the cube root.

The author thus recognizes the embedding of root-taking, and transfers this to the name censo di cubo, corresponding to our (x3)2; this, however, does not force him to give up the multiplicative name for the ﬁfth power. Radice relata, we observe, coincides with Antonio’s name for the same power.

61 Benedetto as well as Palat. 573, both of which copy long extracts from Antonio, also take over his naming in their independent chapters.

The third encyclopedia instead uses both cubo di censo and censo di cubo about the ﬁfth power; the intervening 70 years have thus not witnessed any steps beyond the inconsistencies of the late 14th century.

62 There is some – but not really exhaustive – change toward the end of the century.

The Modena manuscript Bibl. Estense, ital. 578 uses the “root names” for the powers, and the abbreviations C, Z and Q for cosa, censo (thought of in the North Italian orthography zenso) and cubo.

63 The whole sequence is then abbreviated N (power zero) C (ﬁrst power) Z (second power) Q (third power) ZZ (fourth power) Cd¯ZZ (ﬁfth power) ZdiQ (sixth power) Cd¯ZdQ (seventh power) Zd¯ZZ (eighth power) Qd¯Q (ninth power)

64 Since, as always, 2+2 = 2 2, we cannot decide the principle according to which the name for fourth power is formed; the ﬁfth, however, is clearly formed from the fourth as a multiplication, whereas the 6th is based on embedding.

The seventh is based on mixed principles, the eighth and the ninth on pure embedding.

65 Next, a new scheme gives the corresponding root signiﬁcations: C: egli che trovi. Z: la R. di quello. Q: la R. quba di quello. ZÇ: la R. di R. di quello. Cd¯ZZ: la sua R. di quello. ZdiQ: la sua R. de la R. di quello. Cd¯ZdQ: la 7a R. di quello. Zd¯ZZ: la R. di R. di R. di quello. Qd¯Q: la RQ di la RQ di quello. As we see, there is a strong coupling between the roots that are expressed via embedding and the corresponding powers.

The seventh, irreducible root is referred to with this name, whereas the ﬁfth root is unspeciﬁed.

66 All in all, a preliminary conclusion suggests itself: Namely that the much more obvious embedding of roots is what started enforcing also the view of power-taking as an embedding (or, in modern mathematical terms, as a function or an operation).

67 Raffaello Canacci’s Ragionamenti d’algebra from c. 1495 has idiosyncratic names for some of the higher powers: numero (power zero) chosa (ﬁrst power) censo (second power) cubo (third power) censo di censo (fourth power) chubo di censo (ﬁfth power) relato (sixth power) promico (seventh power) censo di censo di censo (eighth power) chubi di chubi (ninth power) relato di censo (tenth power)

68 The ﬁfth power is thus named according to the multiplicative principle, but the eighth and ninth by embedding.

The name for the sixth power is the one others use for the ﬁfth power, and that for the seventh is even more astonishing, and in absolute conﬂict with normal usage.

The name for the tenth power falls outside both systems (but we shall soon see why).

Canacci also experiments with graphic notations for the powers – censo is a square, cubo a vertically divided rectangle, censo di censo two separate squares, his relato a horizontally divided rectangle, his promico a horizontally divided square. Their compositions emulate those of the names.

69 According to Francesco Ghaligai (1521), the same names and graphic signs had been used by Giovanni del Sodo (Canacci’s teacher) in his algebra, with the extension that the 11th power was tromico, and the 13th was dromico.

But del Sodo, according to Ghaligai, used relato about the ﬁfth power, and named the sixth power with embedding, as cubo di censo.

In this system, relato di censo, understood as embedding, is really the tenth power.

Del Sodo’s system is thus consistently based on embedding, although his graphic notation must be characterized as unhandy.

70 Canacci’s inconsistencies, it turns out, must be traced back to deﬁcient understanding of his model and nothing more.

Still, this misunderstanding on the part of an abbacus teacher who was no fool shows his teacher’s insights were not yet commonplace nor central to mathematical thought.

71 In Pacioli’s Perugia manuscript from 1478, those 25 sheets are missing where a systematic presentation of the powers would be expected (also according to Pacioli’s own table of contents).

Since the problems do not deal with powers above the fourth, we can only see that cosa, censo, cubo and censo di censo are represented by superscript co,,Δ and . from which we can derive nothing.

72 His printed Summa from 1494 uses a different notation (plausibly because superscripts were not possible for his printer).

73 Fol. 67v lists the 30 gradi of the “algebraic characters” or dignità (as he says they are called): 1a no. numero (power zero) 2a co. cosa (ﬁrst power) 3a ce. censo (second power) 4a cu. cubo (third power) 5a ce.ce. censo de censo (fourth power) 6a po.ro primo relato (ﬁfth power) 7a ce.cu. censo de cubo e anche cube de censo (sixth power) 8a 2o.r0. secundo relato (seventh power) 9a ce.ce.ce. censo de censo de censo (eighth power) ... 29a ce.ce.2o.ro. censo de censo de secundo relato (twenty-eighth power) 30a [9o]ro. nono relato (twenty-ninth power) Everywhere, composition means embedding, and the prime powers are designated as 1st, 2nd, 3rd, 4th, ... 9th relato.

74 So, with del Sodo and Pacioli, embedding-composition has become the sole principle.

In other words, taking a power has become an operation, and the power itself more or less a function. And in the Modena manuscript as well as Pacioli, the members of the sequence are identiﬁed arithmetically.

75 Chuquet’s Triparty des nombres from 1584 was more radical.

Dropping all names, Chuquet simply wrote the exponent of the power superscript after the coefficient.

This was too radical, at least in the opinion of Étienne de la Roche, whose Larismethique nouvellement composee from 1520, based to a large extent on Chuquet’s work and the only channel through which Chuquet’s ideas reached the world, turns instead to the notations that were becoming current in German algebra at the time, for instance in Rudolff’s Coss (ultimately going back to the Florentine notations of the mid-15th century).

76 Rudolff offers this sequence (omitting the graphic symbols): dragma oder numerus (power zero) radix (ﬁrst power) zensus (second power) cubus (third power) zensdezens (fourth power) sursolidum (ﬁfth power) zensicubus (sixth power) bissursolidum (seventh power) zenszensdezens (eighth power) cubus de cubo (ninth power) – also based on embedding, but with new terms for the prime powers,obviously constructed in a Latinizing environment (sursolidum/supersolidum seems to be related to Antonio’s cubo relato – a cube, after all, is a solid).

77 The same powers and graphic symbols are given by Stifel in the Arithmetica integra – but Stifel goes on until the 16th power (after zensocubicus only with graphic symbols).

Similarly, Tartaglia, in the Secunda parte del general trattato de numeri e misure (1556) repeats Pacioli’s list – and again in the Sesta parte (1560), though stopping now at the 14th power because one very rarely needs so high powers.

78 Bombelli, in the manuscript of his L’algebra, uses an arithmeticized notation with indication of the power written above the coefficient – for instance, for “30 cose”, which in the printed version would become 30 1.

In the beginning of book I, however, he explains the terms which we know from Pacioli and Tartaglia – though only until numero quadro- cubico, over cubicoquadrato.

As Tartaglia in the Sesta parte he obviously sees no purpose in discussing, for the sole reason that they can be given a name, powers that are of no use in his work.

79 All in all, the insights in this domain that had been reached by del Sodo and Pacioli in the late ﬁfteenth century were conserved and systematized but not superseded during the following century.

How could they be, without an explicit parenthesis function, allowing the automatization and trivialization of such insights as xmn =(xm)n = (xn)m ?

Therefore, we shall now turn our attention to the algebraic parenthesis.

80 The parenthesis before and until the brackets

Once upon a time there was a “Babylonian algebra”.

It was discovered (or invented) around 1930, but over the last three decades I believe I have managed to convinced most of those who work seriously on the topic that the numbers found on the tablets and supposed to reﬂect algebraic operations instead correspond to the measures of geometric entities manipulated in a cut-and-paste technique.

81 For instance, let us look at a literal translation of the very simplest second-degree example – the ﬁrst problem on the tablet BM 13901, a “theme text” about squares: 3 1. The surface and my confrontation I have heaped: /4 is it. 1, the projection, 1 1 2. you posit. The moiety of 1 you break, /2 and /2 you make hold. 1 3 1 3. /4 to /4 you join: by 1, 1 is equal. /2 which you have made hold 1 4. from the inside of 1 you tear out: /2 the confrontation.

82 A “confrontation” is the side of a square (it “confronts” its equal), the “moiety” is a “natural half”, a half whose role could not be ﬁlled by any other fraction.

To “make a and b hold” stands for the construction of a rectangle with sides a and b, that s “is equal by” A means that s is the side of the area A laid out as a square.

83 The “projection” gives the clue to the method: at ﬁrst the side or “confrontation” c is provided with a “projection”, a breadth 1, which transforms it into a rectangle with area 1 c = c. Then, according to the statement, this rectangle, 3 together with the square (c), has a total area /4. Breaking it into two equal parts and moving one of 3 them around we get a gnomon, still with area /4, 1 1 1 which is completed by a square of area /2 /2 = /4. 3 1 The completed square has an area /4+ /4 = 1 and therefore a side √1=1. Removal of the part which was added below leaves us 1 1 with the original side, which must hence be 1– /2 = /2.

84 This technique seems to leave no space for anything like a parenthesis, and at this level this immediate impression holds true.

However, the technique may be used for “representation”, that is, the sides of square and rectangular areas may themselves be areas, volumes, numbers of working days or bricks produced during these days, prices, etc.

In the particular case where the sides of a rectangle are two square areas, we may see the solution as making use of an implicit parenthesis – as when Fibonacci solves a bi-biquadratic problem by treating the census census census census as a square area and the census census as its side.

85 The Babylonian texts also present us with en explicit parenthesis function, though only used in very speciﬁc contexts.

They make use of two different subtractive operations, removal and comparison. An entity b can only be removed from another entity B if b is a part of B.

If this is not the case, it may be told by how much B exceeds b.

In the former case, the operation produces an entity that can be subjected to the usual geometric operations.

In the latter, however, only few texts see the excess as an independent quantity that can be directly manipulated, for instance by constructing a square with the excess as side (making the excess “confront itself”).

86 The majority would make “so much as that by which B exceeds A” confront itself. The phrase “so much as” (mala, a single word) thus deﬁnes a parenthesis; it is also used to tell that one entity is “so much as” a composite expressions (for example, “So much as I have made confront itself, and 1 cubit exceeding, that is the depth”, namely of an excavation with square base). However, the use of this parenthesis is not general, and like implicit parentheses (Babylonian, or Fibonacci’s) it cannot be nested without strain on thought. Within the kind of mathematics that was practised (by the Babylonian calculators, or by Fibonacci) it is also dubious whether any use for such nesting would easily present itself.

87 Because of the tails with which the superscript symbols for powers and root were provided in the Maghreb notation, these symbols may serve to delimit parentheses.

The argument of a root sign may be a complex expression, and may even itself contain roots (nesting).

Inverse taking may have an algebraic monomial as its argument, and it may even be repeated; but the notation is ambiguous as regards the coefficient (will it produce 5x–1 or (5x)–1 ?).

Division written fraction-wise may contain algebraic polynomials in the numerator as well as the denominator.

88 The situation concerning the symbols for powers is different. Here, the argument may be an integer or a broken number or even an arithmetical composite, but nothing else.

Number, šai , ma¯l and ka b have individual signs; higher powers may be written as composites either horizontally or vertically, but the meaning will always be multiplicative (as in al-Karajı¯’s verbal list of their names)

– the signs for ma¯l and ka b written together will always mean ma¯l ka b (the ﬁfth power), and not (x3)2.

In other words: šai , ma¯l and ka b are entities, not functions or operations.

89 All in all: at least in its mature phase the Maghreb notation comprised a fairly well developed parenthesis function – certainly more fully developed than anything that can be found in Europe before and even including Viète; but like Viète it stopped short of the point where it could be used for free symbolic manipulation.

90 In Latin (that is, Romance and Germanic) Europe, as we have seen, powers remained entities until the mid-15th century; even with del Sodo, Pacioli etc., when the naming of higher powers was consistently made by embedding, it was still impossible to use their names as operations on other entities than powers of the unknown.

91 Formal fractions carrying a binomial in the denominator were in use from the mid-14th century, as we have seen; in the 15th century, trinomials also appear occasionally.

For roots, the sign came in use before 1340 (Giovanni di Davizzo used it in 1339, and Dardi in 1344).

The cube root, however, was written cubo, roots of composite expressions also had to be designated –“de zonto” (Dardi) – (Gilio, who may have taken it over from his master Antonio,

and also Benedetto – and legata or u (u for universale or unita) by Pacioli and Cardano.

92 Mostly, but not necessarily, this root was to be taken of a binomial; Cardano, moreover, might use u of a sum as the sum of the two roots ( u(a+b)=√a+√b).

That is, it is no symbol at all but only an abbreviation, whose meaning must be understood from context (This corresponds to manuscript abbreviations, where a stroke over a vowel might mean that either m or n was to follow, and where the same abbreviation might stand for phisice as well as philosophice).

93 So, in spite of the original access to inspiration from the Maghreb and to the enduring use of the algebraic parenthesis deﬁned by the fraction line, the obvious need for an unambiguous way to take roots of polynomials, that is, for a delimitation of the radicand as a parenthesis, was only answered by Chuquet, who used the simple trick to underline the radicand.

As far as I have noticed, he does not use the notation for other purposes, and it is never nested.

De la Roche may have found the innovation superﬂuous.

94 For the root of binomials, Bombelli would still use Radice legata or Radice universale, as he explains.

Longer radicands (and sometimes also binomials) are delimited by an initial L. and a ﬁnal invertedΓ ; sometimes, the system is nested (but always with each parenthesis being a radicand).

The lack of system indicates that the purpose is disambiguation and nothing more.

Bombelli’s manuscript, however, goes somewhat further: the whole radicand is underlined, and the beginning and the end of the line are marked by vertical strokes.

95 So, where does the general algebraic parenthesis begin?

What about Descartes?

Firstly, of course, Descartes has the modern, long square root , which can also be nested – for instance,

Next, he uses complex expressions involving multiple parentheses, as in this equation (p. 398):

Descartes_equation

96 As we see, the parentheses are not enclosed in pairs of brackets, but written vertically and kept together by a single curly bracket to the right; but that is immaterial.

We also notice that Descartes prefers to write repeated products like yy, even though he writes y3 (etc.), as in this expression:

Descartes_2

but that, again, is a different question.

97 Descartes does not use these parentheses very much, but they are there.

And as Engels states in Dialektik der Natur, “100,000 steam engines [prove the principle] no more than one”; or, at least, in a formulation ascribed to Wilamowitz-Moellendorf, “according to the philologists, once is never, twice is always”.

98 So, after Descartes, the road was open to Euler’s development of the inﬁnite fractional inﬁnite product

Euler_1 as the sum

Euler_2

99 Why? why not?

Why was progress so slow, much more marked by stops than by goes, at times even by regressions? Or, perhaps, why not?

Metaphysical absolute progress is nothing but an illusion, mistaking “the royal road to me” for the road.

Within the broader practice of ocean trade, colonization and warfare, improved mathematical navigation certainly constitutes progress – but from the point of view of the human chattel brought over the Atlantic or dying on the way, the characterization can be disputed.

100 Even Nunez and Dee, however, had little use for algebra when working on navigational techniques.

Until their time, algebra had no social uses outside the environment of those who lived from teaching mathematics.

Mathematics, of course, also has its internal constraints, and those (like Dardi, Antonio and Benedetto) who understood the subject well would not stoop to the false solutions of irreducible cubics and quartics or Giovanni di Davizzo’s advertising of roots as inverse powers.

101 Even they, however, used and developed algebra in view of treating a particular kind of problems, and for this kind of problems they had no need for developing neither symbolic operations nor embedding and parenthesis function.

Personally (but this is already counterfactual history running wild), they might perhaps have enjoyed it if they had been able to foresee that developing such techniques would have enabled them to discover Euler’s theorems about the partition of numbers (or just Descartes geometrical results).

102 However, in the competition for pupils and prestige within the environment of abbacus teaching such things would not have been understood and therefore would not have counted, and in any case it is in the nature of dialectic to react to the situation which is already there – nobody gets the idea of creating tools for the solution of problems which only practice of these tools will eventually create.

103 Moreover, even when Descartes shaped the tools later used by an Euler, he did not and could not foresee what they would make possible.

He shaped them more or less accidentally within his particular context, and had no reason to use them more than he did, preparing a future he did not know about.

104 Already Descartes, however, lived in a mathematical future unknown even to Stifel and his abbacist predecessors.

Like theirs, his mathematical world was one where problems served as challenges, and where the ability to solve problems was the ground for prestige; but the problems were no longer those of repeated travels with gain, finding a purse and sharing its contents, buying a horse in common or finding numbers in given ratio fulfilling conditions corresponding to particular algebraic equations.

105 Descartes, Wallis and their kind were not Humanists – Humanism, in its heyday had never been interested in mathematics.

But as Humanists discovered after 1500, after the catastrophic grand tour d’Italie of the French artillery and after the discovery of the New World, civic utility if restricted to rhetoric and other studia humanitatis was useless, civically and in general; civic utility had to encompass technology and even of mathematical competence (as reﬂected in Hans Holbein’s Ambassadors).

106 107 In consequence, the Greek mathematicians became interesting, and the editiones primae and the ﬁrst translations of the Greek mathematicians (beyond Euclid and the Measurement of the Circle) were produced.

For French Humanists and post-Humanists like Viète, Fermat and Descartes, worthy problems were therefore those inspired by Archimedes, Apollonios and Pappos.

Algebra was still there, known as the art of solving problems.

But it needed to be reshaped (and not only because of its Arabic name); and that was what they did.

108